Apparatus and method for providing energy - bandwidth tradeoff and waveform design in interference and noise

ABSTRACT

A new method for transmitter-receiver design that enhances the desired signal output from the receiver while minimizing the total interference and noise output from the receiver at the desired decision making instant is presented. Further the new design scheme proposed here can be used for transmit signal energy and bandwidth tradeoff. As a result, transmit signal energy can be used to tradeoff for the “premium” signal bandwidth without sacrificing the system performance level in terms of the output Signal to Interference plus Noise power Ratio (SINR). The two designs—the one before and the one after the tradeoff—will result in two different transmitter-receiver pairs that have the same performance level. In many applications such as in telecommunications, since the available bandwidth is at premium, such a tradeoff will result in releasing otherwise unavailable bandwidth at the expense of additional signal energy. The bandwidth so released can be used for other applications or to add additional telecommunication capacity to the system.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The present invention is based upon work supported and/or sponsored bythe Air Force Research Laboratory (AFRL), Rome, N.Y. under contract No.FA8750-06-C-0202

FIELD OF INVENTION

The invention relates to techniques related to a tradeoff betweentransmit signal Energy and its Bandwidth using new transmitter-receiverwaveform design methods that are applicable for radar, sonar andwireless communications.

BACKGROUND OF INVENTION

In the general problem, a desired target is buried in both interferenceand noise. A transmit signal excites both the desired target and theinterference simultaneously. The interference and/or interferences canbe foliage returns in the form of clutter for radar, scattered returnsof the transmit signal from a sea-bottom and different ocean-layers inthe case of sonar, or multipath returns in a communication scene. In allof these cases, like the target return, the interference returns arealso transmit signal dependent, and hence it puts conflicting demands onthe receiver. In general, the receiver input is comprised of targetreturns, interferences and the ever present noise. The goal of thereceiver is to enhance the target returns and simultaneously suppressboth the interference and noise signals. In a detection environment, adecision regarding the presence or absence of a target is made at somespecified instant ωt=t_(o) using output data from a receiver, and henceto maximize detection, the Signal power to average Interference plusNoise Ratio (SINR) at the receiver output can be used as an optimizationgoal. This scheme is illustrated in FIG. 1.

The transmitter output bandwidth can be controlled using a knowntransmitter output filter having a transfer function P₁(ω) (see FIG.2A). A similar filter with transform characteristics P₂(ω) can be usedat a receiver input 22 a shown in FIG. 1, to control the processingbandwidth as well.

The transmit waveform set f(t) at an output 10 a of FIG. 1, can havespatial and temporal components to it each designated for a specificgoal. The simplest situation is that shown in FIG. 2A where a finiteduration waveform f(t) of energy E is to be designed. Thus

$\begin{matrix}{{\int_{0}^{T_{o}}{{{f(t)}}^{2}{t}}} = {E.}} & (1)\end{matrix}$

Usually, transmitter output filter 12 characteristics P₁(ω), such asshown in FIG. 2B, are known and for design purposes, it is best toincorporate the transmitter output filter 12 and the receiver inputfilter (which may be part of receiver 22) along with the target andclutter spectral characteristics.

Let q(t)

Q(ω) represent the target impulse response and its transform. In generalq(t) can be any arbitrary waveform. Thus the modified target thataccounts for the target output filter has transform P₁(ω)Q(ω) etc. In alinear domain setup, the transmit signal f(t) interacts with the targetq(t), or target 14 shown in FIG. 1, to generate the output below(referred to in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci,“Optimum Transmit-Receiver Design in the Presence of Signal-DependentInterference and Channel Noise”, IEEE Transactions on InformationTheory, Vol. 46, No. 2, pp. 577-584, March 2000 and J. R. Guerci and S.U. Pillai, “Theory and Application of Optimum Transmit-Receive Radar,”IEEE International Radar Conference, Alexandria Va., May 2000, pp.705-710):

$\begin{matrix}{{s(t)} = {{{f(t)}*{q(t)}} = {\int_{0}^{T_{o}}{{f(\tau)}{q\left( {t - \tau} \right)}{\tau}}}}} & (2)\end{matrix}$

that represents the desired signal.

The interference returns are usually due to the random scattered returnsof the transmit signal from the environment, and hence can be modeled asa stochastic signal w_(c)(t) that is excited by the transmit signalf(t). If the environment returns are stationary, then the interferencecan be represented by its power spectrum G_(c)(ω). This gives theaverage interference power to be G_(c)(ω)|F(ω)². Finally let n(t)represent the receiver 22 input noise with power spectral densityG_(n)(ω). Thus the receiver input signal at input 22 a equals

r(t)=s(t)+w _(c)(t)*f(t)+n(t),  (3)

and the input interference plus noise power spectrum equals

G ₁(ω)=G _(c)(ω)|F(ω)|² +G _(n)(ω).  (4)

The received signal is presented to the receiver 22 at input 22 a withimpulse response h(t). The simplest receiver is of the noncausal type.

With no restrictions on the receiver 22 of FIG. 1, its output signal atoutput 22 b in FIG. 1, and interference noise components are given by

$\begin{matrix}{{{y_{s}(t)} = {{{s(t)}*{h(t)}} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{S(\omega)}{H(\omega)}^{{j\omega}\; t}{\omega}}}}}}\text{and}} & (5) \\{{y_{n}(t)} = {\left\{ {{{w_{c}(t)}*{f(t)}} + {n(t)}} \right\}*{{h(t)}.}}} & (6)\end{matrix}$

The output y_(n)(t) represents a second order stationary stochasticprocess with power spectrum below (referred to in the previouspublications and in Athanasios Papoulis, S. Unnikrishna Pillai,Probability, Random Variables and Stochastic Processes, McGraw-HillHigher Education, New York 2002):

G _(o)(ω)=(G _(c)(ω)|F(ω)|² +G _(n)(ω)|H(ω)²  (7)

and hence the total output interference plus noise power is given by

$\begin{matrix}\begin{matrix}{\sigma_{I + N}^{2} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{G_{0}(\omega)}{\omega}}}}} \\{= {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\left( {{{G_{C}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}} \right){{H(\omega)}}^{2}{{\omega}.}}}}}\end{matrix} & (8)\end{matrix}$

Referring back to FIG. 1, the signal component y_(s)(t) in equation (5)at the receiver output 22 b needs to be maximized at the decisioninstant to in presence of the above interference and noise. Hence theinstantaneous output signal power at t=t_(o) is given by the formulabelow shown in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci,“Optimum Transmit-Receiver Design in the Presence of Signal-DependentInterference and Channel Noise”, IEEE Transactions on InformationTheory, Vol. 46, No. 2, pp. 577-584, March 2000, which is incorporatedby reference herein:

$\begin{matrix}{P_{O} = {{{y_{s}\left( t_{o} \right)}}^{2} = {{{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{S(\omega)}{H(\omega)}^{{j\omega}\; t_{o}}{\omega}}}}}^{2}.}}} & (9)\end{matrix}$

This gives the receiver output SINR at t=t_(o) be the following asspecified in Guerci et. al., “Theory and Application of OptimumTransmit-Receive Radar”, pp. 705-710; and Pillai et. al., “OptimumTransmit-Receiver Design in the Presence of Signal-DependentInterference and Channel Noise”, incorporated herein by reference:

$\begin{matrix}{{SINR} = {\frac{P_{o}}{\sigma_{I + N}^{2}} = {\frac{{{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{S(\omega)}{H(\omega)}^{{j\omega}\; t_{o}}{\omega}}}}}^{2}}{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{G_{I}(\omega)}{{H(\omega)}}^{2}{\omega}}}}.}}} & (10)\end{matrix}$

We can apply Cauchy-Schwarz inequality in equation (10) to eliminateH(ω). This gives

$\begin{matrix}\begin{matrix}{{SINR} \leq {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{{{S(\omega)}}^{2}}{G_{I}(\omega)}{\omega}}}}} \\{= {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{{{Q(\omega)}}^{2}{{F(\omega)}}^{2}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}{\omega}}}}} \\{= {{SINR}_{\max}.}}\end{matrix} & (11)\end{matrix}$

Thus the maximum obtainable SINR is given by equation (11), and this isachieved if and only if the following equation referred to in previousprior art publications, is true:

$\begin{matrix}\begin{matrix}{{H_{opt}(\omega)} = {\frac{S*(\omega)}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}^{{- {j\omega}}\; t_{o}}}} \\{= {\frac{Q*(\omega)F*(\omega)}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}{^{{- {j\omega}}\; t_{o}}.}}}\end{matrix} & (12)\end{matrix}$

In (12), the phase shift e^(−jωt) ^(o) can be retained to approximatecausality for the receiver waveform. Interestingly even with a pointtarget (Q(ω)≡1), flat noise (G_(n)(ω)=σ_(n) ²), and flat clutter(G_(c)(ω)=σ_(c) ²), the optimum receiver is not conjugate-matched to thetransmit signal, since in that case from equation (12) we have thefollowing formula given by Pillai et. al., “Optimum Transmit-ReceiverDesign in the Presence of Signal-Dependent Interference and ChannelNoise”, incorporated herein by reference, Papoulis, “Probability, RandomVariables and Stochastic Processes”, and H. L. Van Trees, Detection,Estimation, and Modulation Theory, Part I, New York: John Wiley andSons, 1968, incorporated by reference:

$\begin{matrix}{{H_{opt}(\omega)} = {{\frac{F*(\omega)}{{\sigma_{c}^{2}{{F(\omega)}}^{2}} + \sigma_{n}^{2}}^{{- {j\omega}}\; t_{o}}} \neq {F*(\omega){^{{- {j\omega}}\; t_{o}}.}}}} & (13)\end{matrix}$

Prior Art Transmitter Waveform Design

When the receiver design satisfies equation (12), the output SINR isgiven by the right side of the equation (11), where the free parameter|F(ω)|² can be chosen to further maximize the output SINR, subject tothe transmit energy constraint in (1). Thus the transmit signal designreduces to the following optimization problem:

Maximize

$\begin{matrix}{{{SINR}_{\max} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{{{Q(\omega)}}^{2}{{F(\omega)}}^{2}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}{\omega}}}}},} & (14)\end{matrix}$

subject to the energy constraint

$\begin{matrix}{{\int_{0}^{T_{o}}{{{f(t)}}^{2}{t}}} = {{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{{F(\omega)}}^{2}{\omega}}}} = {E.}}} & (15)\end{matrix}$

To solve this new constrained optimization problem, combine (14)-(15) todefine the modified Lagrange optimization function (referred to in T.Kooij, “Optimum Signal in Noise and Reverberation”, Proceeding of theNATO Advanced Study Institute on Signal Processing with Emphasis onUnderwater Acoustics, Vol. I, Enschede, The Netherlands, 1968.)

$\begin{matrix}{{\Lambda = {\int_{- \infty}^{+ \infty}{\left\{ {\frac{{{Q(\omega)}}^{2}{y^{2}(\omega)}}{{{G_{c}(\omega)}{y^{2}(\omega)}} + {G_{n}(\omega)}} - {\frac{1}{\lambda^{2}}{y^{2}(\omega)}}} \right\} {\omega}}}}\text{where}} & (16) \\{{y(\omega)} = {{F(\omega)}}} & (17)\end{matrix}$

is the free design parameter. From (16) (17),

$\frac{\partial\Lambda}{\partial y} = {0\mspace{14mu} {gives}}$

gives

$\begin{matrix}{\frac{\partial{\Lambda (\omega)}}{\partial y} = {{2{y(\omega)}\left\{ {\frac{{G_{n}(\omega)}{{Q(\omega)}}^{2}}{\left\{ {{{G_{c}(\omega)}y^{2}} - (\omega) - {G_{n}(\omega)}} \right\}^{2}} - \frac{1}{\lambda^{2}}} \right\}} = 0.}} & (18)\end{matrix}$

where Λ(ω) represents the quantity within the integral in (16). From(18), either

$\begin{matrix}{{y(\omega)} = 0} & (19) \\{or} & \; \\{{{\frac{{G_{n}(\omega)}{{Q(\omega)}}^{2}}{\left\{ {{{G_{c}(\omega)}{y^{2}(\omega)}} + {G_{n}(\omega)}} \right\}^{2}} - \frac{1}{\lambda^{2}}} = 0},} & (20)\end{matrix}$

which gives

$\begin{matrix}{{y^{2}(\omega)} - \frac{\sqrt{G_{n}(\omega)}\left( {{\lambda {{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)}} & (21)\end{matrix}$

provided y²(ω)>0. See T. Kooij cited above incorporated by referenceherein.

SUMMARY OF THE INVENTION

One or more embodiments of the present invention provide a method and anapparatus for transmitter-receiver design that enhances the desiredsignal output from the receiver while minimizing the total interferenceand noise output at the desired decision making instant. Further themethod and apparatus of an embodiment of the present invention can beused for transmit signal energy-bandwidth tradeoff. As a result,transmit signal energy can be used to tradeoff for “premium” signalbandwidth without sacrificing performance level in terms of the outputSignal to Interference plus Noise power Ratio (SINR). The twodesigns—before and after the tradeoff—will result in two differenttransmitter-receiver pairs that have the same performance level. Thus adesign that uses a certain energy and bandwidth can be traded off with anew design that uses more energy and lesser bandwidth compared to theold design. In many applications such as in telecommunications, sincethe available bandwidth is at premium, such a tradeoff will result inreleasing otherwise unavailable bandwidth at the expense of additionalsignal energy. The bandwidth so released can be used for otherapplications or to add additional telecommunications capacity.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a diagram of a system, apparatus, and/or method including atransmitter, a transmitter output filter, a receiver, a target,interference, noise, and a switch;

FIG. 2A shows a prior art graph of a prior art transmitter signal versustime, wherein the transmitter signal is output from a transmitter, suchas in FIG. 1;

FIG. 2B shows a prior art graph of a possible frequency spectrum of aknown transmitter output filter, such as in FIG. 1;

FIG. 3A shows a graph of target transfer function magnitude responseversus frequency;

FIG. 3B shows a graph of target transfer function magnitude responseversus frequency;

FIG. 3C shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency;

FIG. 3D shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency;

FIG. 4A shows graphs of three different target transfer functionmagnitude responses versus frequency;

FIG. 4B shows a graph of noise power spectrum versus frequency;

FIG. 4C shows a graph of clutter power spectrum versus frequency;

FIG. 4D shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency;

FIG. 4E shows a graph of transmitter threshold energy versus bandwidth;

FIG. 4F shows a graph of signal to inference plus noise ratio (SINR)versus bandwidth;

FIG. 5A shows graphs of three different target transfer functionmagnitude responses versus frequency;

FIG. 5B shows a graph of noise power spectrum versus frequency;

FIG. 5C shows a graph of clutter power spectrum versus frequency;

FIG. 5D shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency;

FIG. 5E shows a graph of transmitter threshold energy versus bandwidth;

FIG. 5F shows a graph of signal to inference plus noise ratio (SINR)versus bandwidth;

FIG. 6A shows a graph of signal to interference plus noise ratio versusenergy for a resonant target shown in FIG. 5A (solid line);

FIG. 6B shows a graph of signal to interference plus noise ratio versusenergy for a low pass target shown in FIG. 5A (dashed line);

FIG. 6C shows a graph of signal to interference plus noise ratio versusenergy for a flat target shown in FIG. 5A (dotted line);

FIG. 7 shows a graph of signal to interference plus noise ratio versusenergy and the Bandwidth-Energy swapping design;

FIG. 8A shows a graph of the transform of the transmitter signal versusfrequency corresponding to the design point A in FIG. 7;

FIG. 8B shows a graph of the transform of the transmitter signal versusfrequency corresponding to the design point B in FIG. 7;

FIG. 8C shows a graph of the transform of the transmitter signal versusfrequency corresponding to the design point C in FIG. 7; and

FIG. 9 is a graph of realizable bandwidth savings versus operatingbandwidth.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a diagram of a system, apparatus, and/or method 1,including a transmitter 10, a transmitter output filter 12, a target 14,interference 16, noise 18, a summation block 20, receiver 22, and aswitch 24. The present invention, in one or more embodiments, provides anew method and apparatus, by selecting a particular transmit signalf(t), to be output from transmitter 10, and a type of receiver orreceiver transfer function for receiver 22 in accordance with criteriato be discussed below.

The transmitter 10 transmits an output signal f(t) at its output 10 aand supplies this signal to the transmitter output filter 12. Asremarked earlier, for design purposes, the transmitter output filter 12can be lumped together with the target transfer function as well as theinterference spectrum. The transmit signal f(t) passes through theairwaves and interacts with a target 14 and interference 16. Thetarget-modified as well as the clutter-modified (or interferencemodified) versions of the transmit signal f(t) are supplied to thesummation block 18 along with receiver noise 18. The summation block 18may simply be used for description purposes to indicate that the targetmodified, clutter modified, and noise signals combine together. Acombination signal is supplied to receiver 22 at its input 22 a. Thereceiver 22 applies a transfer function H(ω) (which will be determinedand/or selected by criteria of an embodiment of the present invention,to be described below) and a modified combination signal is provided ata receiver output 22 b. The output is accessed at time t=t₀ by use ofswitch 24.

FIG. 2A shows a prior art graph of a prior art transmitter output signalf(t) versus time. The signal used here is arbitrary.

FIG. 2B shows a prior art graph of a frequency spectrum of thetransmitter output filter 12 of FIG. 1.

FIG. 3A shows a typical graph of a target transfer function magnituderesponse for target 14 versus frequency; target as appearing in(14)-(21).

FIG. 3B shows a typical graph of target transfer function magnituderesponse for target 14 versus frequency; target as appearing in(14)-(21).

FIG. 3C shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency; as in right side ofequation (23).

FIG. 3D shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency; as in right side ofequation (23).

FIG. 4A shows graphs of three different target transfer functionmagnitude responses versus frequency; target as appearing in (14)-(21).

FIG. 4B shows a graph of noise power spectrum versus frequency asappearing in equations (14)-(23).

FIG. 4C shows a graph of clutter power spectrum versus frequency asappearing in equations (14)-(23).

FIG. 4D shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency as in right side ofequation (23).

FIG. 4E shows a graph of transmitter threshold energy versus bandwidthusing equation (26).

FIG. 4F shows a graph of signal to inference plus noise ratio versusbandwidth using equations (27)-(31).

FIG. 5A shows graphs of three different target transfer functionmagnitude responses versus frequency; target as appearing in (14)-(21).

FIG. 5B shows a graph of noise power spectrum versus frequency asappearing in equations (14)-(23).

FIG. 5C shows a graph of clutter power spectrum versus frequency asappearing in equations (14)-(23).

FIG. 5D shows a graph of noise power spectrum divided by target transferfunction magnitude response versus frequency as in right side ofequation (23).

FIG. 5E shows a graph of transmitter threshold energy versus bandwidthusing equation (26).

FIG. 5F shows a graph of signal to inference plus noise ratio versusbandwidth using equations (27)-(31).

FIG. 6A shows a graph of signal to interference plus noise ratio versusenergy for a resonant target shown in FIG. 5A (solid line) usingequations (34)-(35).

FIG. 6B shows a graph of signal to interference plus noise ratio versusenergy for a low pass target shown in FIG. 5A (dashed line) usingequations (34)-(35).

FIG. 6C shows a graph of signal to interference plus noise ratio versusenergy for a flat target shown in FIG. 5A (dotted line) using equations(34)-(35).

FIG. 7 shows a graph of signal to interference plus noise ratio versusenergy; generated using equations (39), (48), and (51).

FIG. 8A shows a graph of the transform of the transmitter signal versusfrequency corresponding to the design point A in FIG. 7 generated using(42).

FIG. 8B shows a graph of the transform of the transmitter signal versusfrequency corresponding to the design point B in FIG. 7 generated using(42).

FIG. 8C shows a graph of the transform of the transmitter signal versusfrequency corresponding to the design point C in FIG. 7 generated using(42) for a third energy condition.

FIG. 9 is a realizable bandwidth savings versus operating bandwidthgenerated using equation (60).

Define Ω₊ as shown in FIGS. 3C and 3D to represent the frequencies overwhich y²(ω) in equation (21) is strictly positive, and let ω_(o)represent the complement of Ω₊. As shown in FIGS. 3C and 3D, observethat the set Ω₊ is a function of the noise and target spectralcharacteristics as well as the constraint constant λ. In terms of Ω₊, wehave

$\begin{matrix}{{{F(\omega)}}^{2} = \left\{ \begin{matrix}{{y^{2}(\omega)},} & {\omega \in \Omega_{+}} \\{0,} & {\omega \in {\Omega_{o}.}}\end{matrix} \right.} & (22)\end{matrix}$

From (21), y²(ω)>0 over Ω₊ gives the necessary condition

$\begin{matrix}{\lambda \geq {\max\limits_{\omega \in \Omega_{+}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}} & (23)\end{matrix}$

and the energy constraint in (15) when applied to (21) gives

$\begin{matrix}{E = {{\frac{1}{2\pi}{\int_{\Omega_{+}}{{y^{2}(\omega)}{\omega}}}} = {{\frac{\lambda}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\pi}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}}} & (24)\end{matrix}$

or, for a given value of E, we have

$\begin{matrix}{\lambda = {\frac{E + {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}}{{{\bullet\lambda}(E)}.}}} & (25)\end{matrix}$

Clearly, λ(E) in (25) must satisfy the inequality in (23) as well. Thisgives rise to the concept of transmitter energy threshold that ischaracteristic to this design approach.

Transmitter Threshold Energy

From (23)-(25), the transmit energy E must be such that λ(E) obtainedfrom (25) should satisfy (23). If not, E must be increased toaccommodate it, and hence it follows that there exists a minimumthreshold value for the transmit energy below which it will not bepossible to maintain |F(ω)|²>0. This threshold value is given by

$\begin{matrix}{E_{\min} = {{\left( {\max\limits_{\omega \in \Omega_{-}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}} & (26)\end{matrix}$

and for any operating condition, the transmit energy E must exceedE_(min). Clearly, the minimum threshold energy depends on the target,clutter and noise characteristics as well as the bandwidth underconsideration. With E>E_(min), substituting (20)-(21) into theSINR_(max) in (14) we get

$\begin{matrix}\begin{matrix}{{SINR}_{\max} = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}^{2}{y^{2}(\omega)}}{\lambda \sqrt{G_{n}(\omega)}{{Q(\omega)}}}{\omega}}}}} \\{= {\frac{1}{2\pi}{\int_{\Omega_{-}}\frac{{Q(\omega)}}{{\lambda (E)}\sqrt{G_{n}(\omega)}}}}} \\{{\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda (E)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)}{\omega}}} \\{= {\frac{1}{2\pi}{\int_{\Omega_{-}}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda (E)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{{\omega}.}}}}}\end{matrix} & (27)\end{matrix}$

Finally making use of (25), the output SINR_(max) becomes

$\begin{matrix}\begin{matrix}{{SINR}_{1} = {{\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\omega}}}} - \frac{\left( {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}\sqrt{G_{n}(\omega)}}{G_{c}(\omega)}{\omega}}}} \right)^{2}}{E + {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}}} \\{= {a - \frac{c}{\lambda (E)}}} \\{= {a - \frac{c^{2}}{E + b}}} \\{= \frac{{aE} + \left( {{ab} - c^{2}} \right)}{E + b}}\end{matrix} & (28) \\{where} & \; \\{{a = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\omega}}}}},} & (29) \\{{b = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}},} & (30) \\{and} & \; \\{c = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{{\omega}.}}}}} & (31)\end{matrix}$

Notice that ab−c²≧0. (This was published in Waveform Diversity andDesign conference, Kauai, Hi., January 2006).

The optimization problem in (14)-(15) can be restated in term of Ω₊ asfollows: Given Q(ω), G_(c)(ω), G_(n)(ω) and the transmit energy E, howto partition the frequency axis into an “operating band” Ω₊ and a “noshow” band Ω_(o) so that λ₊ obtained from (25) satisfies (23) andSINR_(max) in (27)-(28) is also maximized. In general maximization ofSINR_(max) in (27)-(28) over Ω₊ is a highly nonlinear optimizationproblem for arbitrary Q(ω), G_(c)(ω) and G_(n)(ω).

In what follows a new approach to this problem is presented.

An Embodiment of the Present Invention—Desired Band Approach

One approach in this situation is to make use of the “desired frequencyband” of interest B_(o) this is usually suggested by the target responseQ(ω) (and the transmitter output filter) to determine the operating bandΩ₊. The desired band B_(o) can represent a fraction of the totalavailable bandwidth, or the whole bandwidth itself. The procedure fordetermining Ω₊ is illustrated in FIGS. 3A-3C and FIGS. 3B-3D for twodifferent situations. In FIGS. 3A-3D, the frequency band B_(o)represents the desired band, and because of the nature of the noise andclutter spectra, it may be necessary to operate on a larger region Ω₊ inthe frequency domain. Thus the desired band B_(o) is contained alwayswithin the operating band Ω₊. To determine Ω₊, using equation (23) weproject the band B_(o) onto the spectrum √{square root over(G_(n)(ω))}/|Q(ω)| and draw a horizontal line corresponding to

$\begin{matrix}{\lambda_{B_{o}} = {\max\limits_{\omega \in B_{o}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}} & (32)\end{matrix}$

as shown there. Define Ω₊(B_(o)) to represent the frequency region where

$\begin{matrix}{{{\omega \in {\Omega_{+}\left( B_{o} \right)}}:{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq \lambda_{B_{o}}}} = {\max\limits_{\omega \in B_{o}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}} & (33)\end{matrix}$

This procedure can give rise to two situations as shown in FIG. 3A andFIG. 3B. In FIG. 3A, the operating band Ω₊(B_(o)) coincides with thedesired band B_(o) as shown in FIG. 3C, whereas in FIG. 3B, the desiredband B_(o) is a subset of Ω₊(B_(o)) as seen from FIG. 3D.

Knowing Ω₊(B_(o)), one can compute λ=λ(E) with the help of equation (25)over that region, and examine whether A so obtained satisfies (23). Ifnot, the transmitter energy E is insufficient to maintain the operatingband Ω₊(B_(o)) given in (33), and either E must be increased, orΩ₊(B_(o)) must be decreased (by decreasing B_(o)) so that (23) issatisfied. Thus for a given desired band B_(o) (or an operating bandΩ₊(B_(o))), as remarked earlier, there exists a minimum transmitterthreshold energy E_(B) _(o) , below which it is impossible to maintain|F(ω)²>0 over that entire operating band.

Threshold Energy

From equations (24) and (32), we obtain the minimum transmitterthreshold energy in this case to be the following

$\begin{matrix}\begin{matrix}{E_{B_{o}} = {{\frac{\lambda_{B_{o}}}{2\pi}{\int_{\Omega_{+}{(B_{o})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}} -}} \\{{\frac{1}{2\pi}{\int_{\Omega_{-}{(B_{o})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}} \\{= {{{\lambda_{B_{o}}c_{o}} - b_{o}} > 0.}}\end{matrix} & (34)\end{matrix}$

With E≧E_(B) _(o) , the SINR_(max) a in (28) can be readily computed. Inparticular with E=E_(B) _(o) , we get

$\begin{matrix}{{SINR}_{1} = {{{SINR}_{1}\left( B_{o} \right)} = {a_{o} - {\frac{c_{o}^{2}}{E_{B_{o}} + b_{o}}.}}}} & (35)\end{matrix}$

Here a_(o), b_(o) and c_(o) are as given in (29)-(31) with Ω₊ replacedby Ω₊(B_(o)). Eq. (35) represents the performance level for bandwidthB_(o) using its minimum threshold energy. From (21), we also obtain theoptimum transmit signal transform corresponding to energy E_(B) _(o) tobe

$\begin{matrix}\begin{matrix}{{{F(\omega)}}^{2} = \left\{ \begin{matrix}{\frac{\sqrt{G_{n}(\omega)}\left( {{\lambda_{B_{o}}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix} \right.} \\{= \left\{ {\begin{matrix}{{\sqrt{G_{n}(\omega)}\begin{pmatrix}{\left( {\max\limits_{\omega \in B_{o}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right) -} \\\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}\end{pmatrix}\frac{{Q(\omega)}}{G_{c}(\omega)}},} & {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix}.} \right.}\end{matrix} & (36)\end{matrix}$

To summarize, to maintain a given desired band B_(o), there exists anoperating band Ω₊(B_(o))≧B_(o) over which |F(ω)|²<0 and to guaranteethis, the transmit energy must exceed a minimum threshold value E_(B)_(o) given by (34).

FIGS. 4A-F shows the transmitter threshold energy E in (34) and thecorresponding SINR in (35) as a function of the desired bandwidth B_(o)for various target, clutter, and noise spectra. Target to noise ratio(TNR) is set at 0 dB, and the clutter to noise power ratio (CNR) is setat 20 dB here. The total noise power is normalized to unity. The desiredbandwidth B_(o) is normalized with respect to the maximum availablebandwidth (e.g., carrier frequency).

In FIGS. 4A-F, the noise and clutter have flat spectra and for thehighly resonant target (solid line), the required minimum energythreshold and the SINR generated using (34)-(35) reach a saturationvalue for small values of the bandwidth. In the case of the other twotargets, additional bandwidth is required to reach the maximumattainable SINR. This is not surprising since for the resonant target, asignificant portion of its energy is concentrated around the resonantfrequency. Hence once the transmit signal bandwidth reaches the resonantfrequency, it latches onto the target features resulting in maximum SINRat a lower bandwidth.

FIGS. 5A-F show results for a new set of clutter and noise spectra asshown there; the transmitter threshold energy E in (34) and thecorresponding SINR in (35) as a function of the desired bandwidth B_(o)show similar performance details.

From FIG. 5F, in the case of the resonant target (solid curve) the SINRreaches its peak value resulting in saturation even when B_(o) is asmall fraction of the available bandwidth. This is because in that case,the transmit waveform is able to latch onto the dominant resonantfrequency of the target. On the other extreme, when the target has flatcharacteristics (dotted curve), there are no distinguishing frequenciesto latch on, and the transmitter is unable to attain the above maximumSINR even when B_(o) coincides with the total available bandwidth. For alow pass target (dashed curve), the transmitter is indeed able todeliver the maximum SINR by making use of all the available bandwidth.

As FIG. 3B shows, Ω₊(B_(o)) can consist of multiple disjoint frequencybands whose complement Ω_(o) represents the “no show” region. Noticethat the “no show” region Ω_(o) in the frequency domain in (36) for theoptimum transmit signal can be controlled by the transmit energy E in(25). By increasing E, these “no show” regions can be made narrower andthis defines a minimum transmitter threshold energy E_(∞) that allowsΩ₊(B_(o)) to be the entire available frequency axis. To determine E_(∞),let λ_(∞) represent the maximum in (23) over the entire frequency axis.Thus

$\begin{matrix}{{\lambda_{\infty} = {\max\limits_{{\omega } < \infty}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},} & (37)\end{matrix}$

and let a_(∞), b_(∞), c_(∞) refer to the constants a, b, c in (29)-(31)calculated with Ω₊ representing the entire frequency axis. Then from(24)

$\begin{matrix}{E_{\infty} = {{{\lambda_{\infty}c_{\infty}} - b_{\infty}} = {{{\frac{\lambda_{\infty}}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}} - {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}} > 0}}} & (38)\end{matrix}$

represents the minimum transmit energy (threshold) required to avoidpartitioning in the frequency domain. With E_(∞) as given by (38), weobtain SINR_(max) to be (use (28))

$\begin{matrix}{{{SINR}_{1}(\infty)} = {{a_{\infty} - \frac{c_{\infty}}{\lambda_{\infty}}} = {{a_{\infty} - \frac{c_{\infty}^{2}}{E_{\infty} + b_{\infty}}} > 0}}} & (39) \\{and} & \; \\{{{{F(\omega)}}^{2} = \frac{\sqrt{G_{n}(\omega)}\left( {{\lambda_{\infty}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)}},{{\omega } < {\infty.}}} & (40)\end{matrix}$

Clearly by further increasing the transmit energy in (39) beyond that in(38) we obtain

$\begin{matrix}{{{SINR}_{1}->a_{\inf}} = {\frac{1}{2\pi}{\int_{- \infty}^{- \infty}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{{\omega}.}}}}} & (41)\end{matrix}$

It follows that to avoid any restrictions in the frequency domain forthe transmit signal, the transmitter energy E must exceed a minimumthreshold value E_(∞) given by (38) and (39) represents the maximumrealizable SNR. By increasing E beyond E_(∞), the performance can beimproved upto that in (41).

In general from (34) for a given desired bandwidth B_(o), the transmitenergy E must exceed its threshold value E_(B) _(o) . With E>E_(B) _(o)and λ(E) as in (25), the corresponding optimum transmit signal transformis given by (see (21) (22))

$\begin{matrix}{{{F(\omega)}}^{2} = \left\{ \begin{matrix}{\frac{\sqrt{G_{c}(\omega)}\left( {{{\lambda (E)}{(\omega)}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix} \right.} & (42)\end{matrix}$

and clearly this signal is different from the minimum threshold energyone in (36). From (28), the performance level SINR₁(E,B_(o))corresponding to (42) is given by (35) with E_(B) _(o) replaced by E.Thus

$\begin{matrix}{{{SINR}_{1}\left( {E,B_{o}} \right)} = {{a_{o} - \frac{c_{o}^{2}}{E + b_{o}}} > {{{SINR}_{1}\left( B_{o} \right)}.}}} & (43)\end{matrix}$

From (43), for a given bandwidth B_(o), performance can be increasedbeyond that in (35) by increasing the transmit energy. Hence it followsthat SINR₁(B_(o)) represents the minimum performance level for bandwidthB_(o) that is obtained by using its minimum threshold energy. It isquite possible that this improved performance SINR₁(E,B_(o)) can beequal to the minimum performance level corresponding to a higherbandwidth B₁>B_(o). This gives rise to the concept of Energy-Bandwidthtradeoff at a certain performance level. Undoubtedly this is quiteuseful when bandwidth is at premium.

FIGS. 5E-5F exhibit the transmit threshold energy and maximum outputSINR₁(B_(o)) as a function of the desired bandwidth B_(o). Combiningthese figures using (35), an SINR vs. transmit threshold energy plot canbe generated as in FIGS. 6A-C for each target situation.

For example, FIG. 6A-C corresponds to the three different targetsituations considered in FIG. 5 with clutter and noise spectra as shownthere. Notice that each point on the SINR-Energy threshold curve foreach target is associated with a specific desired bandwidth. Thus forbandwidth B₁, the minimum threshold energy required is E₁, and thecorresponding SINR equals SINR₁(B₁) in (35). Let A represent theassociated operating point in FIG. 6. Note that the operating point Acorresponding to a bandwidth B₁ has different threshold energies anddifferent performance levels for different targets. From (35), eachoperating point generates a distinct transmit waveform. As the bandwidthincreases, from (39), SINR→SINR₁(∞).

Monotonic Property of SINR

The threshold energy and SINR associated with a higher bandwidth ishigher. To prove this, consider two desired bandwidths B₁ and B₂ withB₂>B₁. Then from (32) we have

$\begin{matrix}{{\lambda_{2} = {{{\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} > \lambda_{1}} = {\max\limits_{\omega \in B_{1}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}}},} & (44)\end{matrix}$

and from FIG. 3, the corresponding operating bandwidths Ω₊(B_(o)) andΩ₊(B₂) satisfy

Ω₊(B ₂)≧Ω₊(B ₁).  (45)

From (34) (or (24)), the minimum threshold energies are given by

$\begin{matrix}{{E_{i} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{i})}}{\sqrt{G_{n}(\omega)}\left( {\lambda_{i} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\omega}}}}},{i = 1},2} & (46)\end{matrix}$

and substituting (44) and (45) into (46) we get

E₂>E₁.  (47)

Also from (27), the performance levels at threshold energy SINR₁(B_(i))equals

$\begin{matrix}{{{SINR}_{1}\left( B_{i} \right)} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{i})}}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda_{i}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\omega}}}}} & (48)\end{matrix}$

and an argument similar to (44)-(45) gives

SINR ₁(B ₂)≧SINR ₁(B ₁)  (49)

for B₂>B₁. Thus as FIGS. 5A-F-FIGS. 6A-C show, SINR₁(B_(i)) is amonotonically nondecreasing function of both bandwidth and energy. FIG.7 illustrates this SINR-energy relation for the target with flatspectrum shown in FIG. 5A. In FIG. 7, the two operating points A and Bare associated with bandwidths B₁ and B₂, threshold energies E₁ and E₂,and performance levels SINR₁(B₁) and SINR₁(B₂) respectively. Since

B₂>B₁

E₂≧E₁ and SINR ₁(B ₂)≧SINR ₁(B ₁).  (50)

The distinct transmit waveforms |F₁(ω)|² and |F₂(ω)|² associated withthese operating point A and B are given by (36) and they are shown inFIGS. 8A and 8B.

Consider the operating point A associated with the desired bandwidth B₁.If the transmit energy E is increased beyond the corresponding thresholdvalue E₁ with bandwidth held constant at B₁, the performance SINR₁(E,B₁) increases beyond that at A since from (43)

$\begin{matrix}{{{SINR}_{1}\left( {E,B_{1}} \right)} = {{{a_{1} - \frac{c_{1}^{2}}{E + b_{1}}} \geq {a_{1} - \frac{c_{1}^{2}}{E_{1} + b_{1}}}} = {{SINR}_{1}\left( B_{1} \right)}}} & (51)\end{matrix}$

and it is upper bounded by a₁. Here a₁ corresponds to the SINRperformance for bandwidth B₁ as the transmit energy E→∞. Note that a₁,B₁ and c₁ are the constants in (29)-(31) with Ω₊ replaced by Ω₊(B₁). Thedashed curve Aa₁ in FIG. 7 represents SINR₁(E, B₁) for various values ofE. From (42), each point on the curve Aa₁ generates a new transmitwaveform as well.

Interestingly the dashed curves in FIG. 7 cannot cross over the optimumperformance (solid) curve SINR(B_(i)). If not, assume the performanceSINR₁(E,B₁) associated with the operating point A crosses overSINR(B_(i)) at some E₁′>E₁. Then from (47), there exists a frequencypoint B₁′>B₁ with threshold energy E₁′ and optimum performanceSINR₁(B₁′). By assumption,

SINR ₁(E ₁ ′,B ₁)>SINR ₁(B ₁′).  (52)

But this is impossible since SINR₁(B₁′) corresponds to the maximum SINRrealizable at bandwidth B₁′ with energy E₁′, and hence performance at alower bandwidth B₁ with the same energy cannot exceed it. Hence (52)cannot be true and we must have

SINR ₁(E ₁ ′,B ₁)≦SINR ₁(B ₁′),  (53)

i.e., the curves Aa₁, Ba₂, etc. does not cross over the optimumperformance curve ABD.

In FIG. 7, assume that the saturation performance value

a ₁ ≧SINR ₁(B ₂),  (54)

i.e., the maximum performance level for bandwidth B₁ is greater than ofequal to the performance level associated with the operating point Bwith a higher bandwidth B₂ and a higher threshold energy E₂. Draw ahorizontal line through B to intersect the curve Aa₁ at C, and drop aperpendicular at C to intersect the x-axis at E₃. From (51) with E=E₃ weget

SINR ₁(E ₃ ,B ₁)=SINR ₁(B ₂).  (55)

Thus the operating point C on the curve Aa₁ is associated with energyE₃, bandwidth B₁ and corresponds to a performance level of SINR₁(B₂)associated with a higher bandwidth. Notice that

E₃>E₂>E₁, and B₁<B₂.  (56)

In other words, by increasing the transmit energy from E₁ to E₃ whileholding the bandwidth constant at B₁, the performance equivalent to ahigher bandwidth B₂ can be realized provided B₂ satisfies (54). As aresult, energy-bandwidth tradeoff is possible within reasonable limits.The transmit waveform |F₃(ω)|² associated with the operating point C isobtained using (42) by replacing E and B_(o) there with E₃ and B₁respectively. and it is illustrated in FIG. 8C. In a similar manner, thewaveforms corresponding to the operating points A and B in FIG. 7 can beobtained using equation (42) by replacing the energy-bandwidth pair(E,B₀) there with (E₁,B₁) and (E₂,B₂) respectively. These waveforms areshown in FIG. 8A and FIG. 8B respectively A comparison with FIGS. 8A and8B show that the waveform at C is different from those associated withoperating point A and B.

It is important to note that although the transmit waveform design|F₃(ω)|² and |F₁(ω)|² correspond to the same bandwidth (with differentenergies E₃ and E₁), one is not a scaled version of the other. Changingtransmit energy from E₁ to E₃ unleashes the whole design procedure andends up in a new waveform |F₃(ω)|² that maintains a performance levelassociated with a larger bandwidth B₂.

The question of how much bandwidth tradeoff can be achieved at anoperating point is an interesting one. From the above argument, equalitycondition in (54) gives the upper bound on how much effective bandwidthincrement can be achieved by increasing the transmit energy. Notice thatfor an operating point A, the desired bandwidth B₁ gives the operatingbandwidth Ω₊(B₁) and from (29) the performance limit

$\begin{matrix}{a_{1} = {\frac{1}{2\pi}{\int_{\Omega_{-}{(B_{1})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\omega}}}}} & (57)\end{matrix}$

for bandwidth B₁ can be computed. Assume B₂>B₁, and from (48) SINR₁(B₂)the minimum performance at B₂ also can be computed, and for maximumbandwidth swapping the nonlinear equation

a ₁ =SINR ₁(B ₂)  (58)

must be solved for B₂. Then

ΔB(B ₁)=B ₂ −B ₁  (59)

represents the maximum bandwidth enhancement that can be realized at B₁.This is illustrated in for the target situation in FIG. 7. Notice thatthe maximum operating bandwidth if finite in any system due to samplingconsiderations and after normalization, it is represented by unity.Hence the upper limit in (59) must be min(1, B₂). This gives

ΔB=min(1,B ₂)−B ₁  (60)

and this explains the linear nature of ΔB for larger value of B_(i). Inthat case, bandwidth can be enhanced by 1−B₁ only.

The design approach described in this section requires the knowledge ofthe target characteristics in addition to the clutter and noise spectra.

Although the invention has been described by reference to particularillustrative embodiments thereof, many changes and modifications of theinvention may become apparent to those skilled in the art withoutdeparting from the spirit and scope of the invention. It is thereforeintended to include within this patent all such changes andmodifications as may reasonably and properly be included within thescope of the present invention's contribution to the art.

1. A method comprising providing a transmitter and a receiver; selectinga desired bandwidth B_(o) for a transmit signal f(t); outputting thetransmit signal f(t) from the transmitter towards a target and towardsinterference; wherein the target produces a target signal; and furthercomprising receiving a combination signal at the receiver, wherein thecombination signal includes noise and the transmit signal f(t) modifiedby interacting with the target and the interference; wherein thereceiver acts on the combination signal to form a receiver outputsignal; wherein the transmit signal f(t) is selected and the receiver isconfigured so that the ratio of the receiver output signal tointerference plus noise power is maximized while maintaining the desiredbandwidth B_(o) for the receiver output signal; and wherein the Fouriertransform F(ω) of the transmit signal f(t) is given by:${{F(\omega)}}^{2} = \left\{ \begin{matrix}{{\sqrt{G_{n}(\omega)}\left( {\left\lbrack {\max\limits_{\omega \in B_{o}}\frac{\sqrt{{G_{n}(\omega)}`}}{{Q(\omega)}}} \right\rbrack - \frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}},} & {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix} \right.$ wherein B_(o) is the desired bandwidth of thetransmit signal f(t); Q(ω) is the target signal Fourier transform;G_(c)(ω) is the interference spectrum; G_(n)(ω) is the noise spectrum;Ω₀ is complementary region in the frequency to Ω₊(B₀) and Q₊(B₀)represents the frequency region where the following inequality issatisfied:$\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{o}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}$2. The method of claim 1 further comprising selecting a receiver filterfor the receiver such that the receiver filter has a Fourier transformH_(opt)(ω) is given by${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F^{*}(\omega)}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}^{{- {j\omega}}\; t_{o}}}$wherein t_(o) is a decision instant at which the target signal is to bedetected.
 3. The method of claim 1 wherein the transmit signal f(t) sogenerated has a minimum energy given by the threshold value E_(min)where$E_{\min} = {{\left( {\max\limits_{\omega \in {\Omega {(B_{0})}}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{0})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{0})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{{\omega}.}}}}}$4. A method comprising providing a transmitter and a receiver; selectinga desired bandwidth B₁ for a transmit signal f(t); selecting a desiredenergy E that exceeds a predetermined energy level E_(min) given below;outputting the transmit signal f(t) from the transmitter towards atarget and towards interference; wherein the target produces a targetsignal; and further comprising receiving a combination signal at thereceiver, wherein the combination signal includes noise and the transmitsignal f(t) modified by interacting with the target and theinterference; wherein the receiver acts on the combination signal toform a receiver output signal; wherein the transmit signal f(t) isselected and the receiver is configured so that the ratio of thereceiver output signal to interference plus noise power is maximizedwhile maintaining the desired bandwidth B₁ for the receiver outputsignal; and wherein${E_{\min} = {{\left( {\max\limits_{\omega \in {\Omega_{+}{(B_{1})}}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}},$and wherein the Fourier transform F(ω) of the transmit signal f(t) isgiven by ${{F(\omega)}}^{2} = \left\{ {{\begin{matrix}{\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda (E)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{1} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix}{wherein}{\lambda (E)}} = {\frac{E + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}}.}} \right.$and wherein B₁ is the desired bandwidth of the transmit signal f(t); Eis the prescribed energy of the transmit signal f (t); Q(ω) is thetarget signal Fourier transform; G_(c)(ω) is the interference spectrum;G_(n)(ω) is the noise spectrum; Ω₀ is complementary region in thefrequency to Ω₊(B_(o)) and Q₊(B₁) represents the frequency region wherethe following inequality is satisfied:$\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{1}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}$5. The method of claim 1 wherein the interference and the noise are partof a space based radar scene.
 6. The method of claim 1 wherein theinterference and the noise are part of an airborne based radar scene. 7.The method of claim 1 wherein the interference and the noise are part ofa ground based radar scene.
 8. The method of claim 1 wherein theinterference and the noise are part of an underwater sonar scene.
 9. Themethod of claim 1 wherein the transmit signal f(t), the target, theinterference, and the noise are part of a cellular communication scene(ground based and/or space based) wherein the transmit signal f(t)represents the desired voice or data modulated signal, the targetrepresents the channel, and the interference represents all interferencesignals.
 10. A method comprising providing a transmitter and a receiver;outputting the transmit signal f(t) from the transmitter towards atarget and towards interference; wherein the target produces a targetsignal; wherein the transmit signal f(t) has a transmit signalbandwidth, transmit signal energy, and a transmit signal waveform; andfurther comprising receiving a combination signal at the receiver,wherein the combination signal includes noise and the transmit signalf(t) modified by interacting with the target and the interference;wherein the receiver acts on the combination signal to form a receiveroutput signal; wherein the receiver output signal has a receiver outputsignal waveform; trading transmit signal bandwidth against transmitsignal energy by redesigning the transmit signal and receiver outputsignal waveforms without sacrificing the performance level.
 11. Themethod of claim 10 further comprising selecting an initial desired firstbandwidth B₁ for the transmit signal bandwidth and determining theminimum required energy E₁ for the transmit signal according to$E_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\sqrt{G_{n}(\omega)}\left( {\lambda_{1} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\omega}}}}$wherein${\lambda_{1} = {\max\limits_{\omega \in B_{1}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},$and determining the performance for this pair at (E₁, B₁) is given by${{SINR}_{1}\left( B_{1} \right)} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda_{1}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{{\omega}.}}}}$wherein Q(ω) is the target signal Fourier transform; G_(c)(ω) is theinterference spectrum; G_(n)(ω) is the noise spectrum; and Ω₊(B₁)represents the frequency region where the following inequality issatisfied:$\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{1}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}$12. The method of claim 11 further comprising selecting a secondbandwidth B₂ larger than B₁ so as to satisfy the condition${{{SINR}_{1}\left( B_{2} \right)} \leq {\frac{1}{2\pi}{\int_{\Omega_{+}B_{1}}^{\;}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\omega}}}}},$and for which the corresponding minimum energy E₂ and performance levelSINR₁(B₂) are determined as follows$E_{2} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{2})}}{\sqrt{G_{n}(\omega)}\left( {\lambda_{2} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\omega}}}}$and${{{SINR}_{1}\left( B_{2} \right)} = {{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{2})}}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda_{2}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{{\omega}.{where}}\mspace{14mu} \lambda_{2}}}} = {\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}}},{and}$Ω₊ (B₂) represents the frequency region where the following inequalityis satisfied:${\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}};$and further comprising determining a new energy level E₃ that satisfiesthe identity${{{SINR}_{1}\left( {E_{3},B_{1}} \right)} = {{a_{1} - \frac{c_{1}^{2}}{E_{3} + b_{1}}} = {{SINR}_{1}\left( B_{2} \right)}}},{where}$${a_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\omega}}}}},{b_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}},{c_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{{\omega}.}}}}}$13. The method of claim 12 further comprising constructing a firsttransmit signal having a first transmit signal waveform; constructing asecond transmit signal having a second transmit signal waveform; whereinthe first transmit signal is different from the second transmit signaland the first transmit signal waveform is different from the secondtransmit signal waveform; wherein the first transmit signal has aprescribed bandwidth which is B₂; wherein the first transmit signal hasan minimum energy which is E₂; wherein the second transmit signal has abandwidth which is B₁; wherein the second transmit signal has aprescribed energy level which is E₃; wherein the first transmit signalhas transform is given by${{F(\omega)}}^{2} = \left\{ {\begin{matrix}{{\sqrt{G_{n}(\omega)}\left( {\left( {\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right) - \frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}},} & {\omega \in {\Omega_{+}\left( B_{2} \right)}} \\{0,} & {otherwise}\end{matrix};} \right.$ wherein the second transmit signal has transformis given by ${{F(\omega)}}^{2} = \left\{ {{\begin{matrix}{\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda \left( E_{3} \right)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{1} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix}{wherein}{\lambda \left( E_{3} \right)}} = {\frac{E_{3} + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}}.}} \right.$and wherein the first and second transmit signals have the sameperformance index SINR₁(B₂) in terms of target detection in interferenceand noise, when used in conjunction with a receiver filter for thereceiver such that the receiver filter has a Fourier transformH_(opt)(ω) given by${H_{opt}(\omega)} = {\frac{Q*(\omega)F*(\omega)}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}^{{- {j\omega}}\; t_{o}}}$wherein t_(o) is a decision instant at which the target signal is to bedetected, and these two waveforms may be substituted for each other. 14.The method of claim 10 wherein the transmit signal f(t), the target, theinterference, and the noise are part of a cellular communication scenewhere the transmit signal f(t) represents the desired voice or datamodulated signal, the target represents the channel, and theinterference represents all interference signals.
 15. The method ofclaim 10 wherein the interference and the noise are part of a spacebased radar scene.
 16. The method of claim 10 wherein the interferenceand the noise are part of an airborne based radar scene.
 17. The methodof claim 10 wherein the interference and the noise are part of a groundbased radar scene.
 18. The method of claim 10 wherein the interferenceand the noise are part of an underwater sonar scene.
 19. An apparatuscomprising a transmitter; and a receiver; wherein the transmitter isconfigured to transmit a transmit signal f(t) of a bandwidth B_(o)towards a target and towards interference; wherein the target produces atarget signal; wherein the receiver is configured to receive acombination signal, wherein the combination signal includes noise andthe transmit signal f(t) modified by interacting with the target and theinterference; wherein the receiver is configured to act on thecombination signal to form a receiver output signal; wherein thetransmit signal f(t) is selected and the receiver is configured so thatthe ratio of the receiver output signal to interference plus noise poweris maximized while maintaining the bandwidth B_(o) for the receiveroutput signal; and wherein the Fourier transform F(ω) of the transmitsignal f(t) is given by: ${{F(\omega)}}^{2} = \left\{ \begin{matrix}{{\sqrt{G_{n}(\omega)}\left( {\left\lbrack {\max\limits_{\omega \in B_{o}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right\rbrack - \frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}},} & {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix} \right.$ wherein B_(o) is the desired bandwidth of thetransmit signal f(t); Q(ω) is the target signal Fourier transform;G_(c)(ω) is the interference spectrum; G_(n)(ω) is the noise spectrum;Ω₀ is complementary region in the frequency to Ω₊(B₀) and Ω₊(B₀)represents the frequency region where the following inequality issatisfied:$\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{o}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}$20. The apparatus of claim 19 further comprising a receiver filter whichis part of the receiver; and wherein the receiver filter is configuredto have a Fourier transform H_(opt)(ω) which is given by${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F^{*}(\omega)}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}^{{- {j\omega}}\; t_{o}}}$wherein t_(o) is a decision instant at which the target signal is to bedetected.
 21. The apparatus of claim 19 wherein the transmit signal f(t)has a minimum energy given by the threshold value E_(min) where$E_{\min} = {{\left( {\max\limits_{\omega \in {\Omega_{+}\; {(B_{0})}}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{1}{2\; \pi}{\int_{\Omega_{+}{(B_{0})}}^{\;}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{0})}}^{\;}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{{\omega}.}}}}}$22. An apparatus comprising a transmitter; and a receiver; wherein thetransmitter is configured to output a transmit signal f(t) having abandwidth B₁ and an energy E that exceeds a predetermined energy levelE_(min) given below; wherein the transmitter is configured to output thetransmit signal f(t) towards a target and towards interference; whereinthe target produces a target signal; wherein the receiver is configuredto receive a combination signal at the receiver, wherein the combinationsignal includes noise and the transmit signal f(t) modified byinteracting with the target and the interference; wherein the receiveris configured to act on the combination signal to form a receiver outputsignal; wherein the transmit signal f(t) is selected and the receiver isconfigured so that the ratio of the receiver output signal tointerference plus noise power is maximized while maintaining thebandwidth B₁ for the receiver output signal; and wherein${E_{\min} = {{\left( {\max\limits_{\omega \in {\Omega_{+}{(B_{1})}}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{1}{2\; \pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}},$and wherein the Fourier transform F(ω) of the transmit signal f(t) isgiven by ${{F(\omega)}}^{2} = \left\{ {{\begin{matrix}{\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda (E)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{1} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix}{wherein}{\lambda (E)}} = {\frac{E + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}}.}} \right.$and wherein B₁ is the desired bandwidth of the transmit signal f(t); Eis the prescribed energy of the transmit signal f(t); Q(ω) is the targetsignal Fourier transform; G_(c)(ω) is the interference spectrum;G_(n)(ω) is the noise spectrum; Ω₀ is complementary region in thefrequency to Ω₊(B_(o)) and Ω₊(B₁) represents the frequency region wherethe following inequality is satisfied$\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{1}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}$23. The apparatus of claim 19 wherein the interference and the noise arepart of a space based radar scene.
 24. The apparatus of claim 19 whereinthe interference and the noise are part of an airborne based radarscene.
 25. The apparatus of claim 19 wherein the interference and thenoise are part of a ground based radar scene.
 26. The apparatus of claim19 wherein the interference and the noise are part of an underwatersonar scene.
 27. The apparatus of claim 19 wherein the transmit signalf(t), the target, the interference, and the noise are part of a cellularcommunication scene (ground based and/or space based) wherein thetransmit signal f(t) represents the desired voice or data modulatedsignal, the target represents the channel, and the interferencerepresents all interference signals.
 28. An apparatus comprising atransmitter; and a receiver; wherein the transmitter is configured tooutput a transmit signal f(t) towards a target and towards interference;wherein the target produces a target signal; wherein the transmit signalf(t) has a transmit signal bandwidth, transmit signal energy, and atransmit signal waveform; wherein the receiver is configured to receivea combination signal at the receiver, wherein the combination signalincludes noise and the transmit signal f(t) modified by interacting withthe target and the interference; wherein the receiver is configured toact on the combination signal to form a receiver output signal; whereinthe receiver output signal has a receiver output signal waveform;wherein the transmitter and the receiver are configured so that transmitsignal bandwidth is traded off against transmit signal energy byredesigning the transmit signal and receiver output signal waveformswithout sacrificing the performance level.
 29. The apparatus of claim 28further comprising wherein the transmitter is configured so that aninitial desired first bandwidth B₁ for the transmit signal bandwidth isselected and a minimum required energy E₁ for the transmit signal isselected according to$E_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\sqrt{G_{n}(\omega)}\left( {\lambda_{1} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\omega}}}}$wherein${\lambda_{1} = {\max\limits_{\omega \in B_{1}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},$and determining the performance for this pair at (E₁, B₁) is given by${{SINR}_{1}\left( B_{1} \right)} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda_{1}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{{\omega}.}}}}$wherein Q(ω) is the target signal Fourier transform; G_(c)(ω) is theinterference spectrum; G_(n)(ω) is the noise spectrum; and Ω₊(B₁)represents the frequency region where the following inequality issatisfied:$\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{1}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}$30. The apparatus of claim 29 further comprising configuring thetransmitter so that a second bandwidth B₂ is selected, which is largerthan B₁ so as to satisfy the condition${{{SINR}_{1}\left( B_{2} \right)} \leq {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\omega}}}}},$and for which the corresponding minimum energy E₂ and performance levelSINR₁(B₂) are determined as follows$E_{2} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{2})}}^{\;}{\sqrt{G_{n}(\omega)}\left( {\lambda_{2} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\omega}}}}$and${{{SINR}_{1}\left( B_{2} \right)} = {{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{2})}}^{\;}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda_{2}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{{\omega}.{where}}\mspace{14mu} \lambda_{2}}}} = {\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}}},{and}$Ω₊(B₂) represents the frequency region where the following inequality issatisfied:${\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}} \leq {\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}};$and wherein the transmitter is configured so that the transmit signalhas an energy level E₃ that satisfies the identity${{{SINR}_{1}\left( {E_{3},B_{1}} \right)} = {{a_{1} - \frac{c_{1}^{2}}{E_{3} + b_{1}}} = {{SINR}_{1}\left( B_{2} \right)}}},{where}$${a_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\omega}}}}},{b_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}},{and}$$c_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{{\omega}.}}}}$31. The apparatus of claim 30 further comprising wherein the transmitteris configured so that it outputs a first transmit signal having a firsttransmit signal waveform; wherein the transmitter is configured so thatit outputs a second transmit signal having a second transmit signalwaveform; wherein the first transmit signal is different from the secondtransmit signal and the first transmit signal waveform is different fromthe second transmit signal waveform; wherein the first transmit signalhas a prescribed bandwidth which is B₂; wherein the first transmitsignal has an minimum energy which is E₂; wherein the second transmitsignal has a bandwidth which is B₁; wherein the second transmit signalhas a prescribed energy level which is E₃; wherein the first transmitsignal has a transform which is given by${{F(\omega)}}^{2} = \left\{ {\begin{matrix}{{\sqrt{G_{n}(\omega)}\left( {\left( {\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right) - \frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}},} & {\omega \in {\Omega_{+}\left( B_{2} \right)}} \\{0,} & {otherwise}\end{matrix};} \right.$ wherein the second transmit signal has atransform which is given by${{F(\omega)}}^{2} = \left\{ {{\begin{matrix}{\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda \left( E_{3} \right)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{1} \right)}} \\{0,} & {\omega \in \Omega_{o}}\end{matrix}{wherein}{\lambda \left( E_{3} \right)}} = {\frac{E_{3} + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}^{\;}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\omega}}}}.}} \right.$and wherein the first and second transmit signals have the sameperformance index SINR₁(B₂) in terms of target detection in interferenceand noise, when used in conjunction with a receiver filter for thereceiver such that the receiver filter has a Fourier transformH_(opt)(ω) given by${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F^{*}(\omega)}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}^{{- {j\omega}}\; t_{o}}}$wherein t_(o) is a decision instant at which the target signal is to bedetected, and these two waveforms may be substituted for each other. 32.The apparatus of claim 28 wherein the transmit signal f(t), the target,the interference, and the noise are part of a cellular communicationscene where the transmit signal f(t) represents the desired voice ordata modulated signal, the target represents the channel, and theinterference represents all interference signals.
 33. The apparatus ofclaim 28 wherein the interference and the noise are part of a spacebased radar scene.
 34. The apparatus of claim 28 wherein theinterference and the noise are part of an airborne based radar scene.35. The apparatus of claim 28 wherein the interference and the noise arepart of a ground based radar scene.
 36. The apparatus of claim 28wherein the interference and the noise are part of an underwater sonarscene.